You are welcome to install third-party applications, but could you clarify how it would be integrated into cPanel/WHM? I'm not familiar with Quickbooks itself, but there are several applications that utilize desktop clients (e.g. FTP clients, MySQL administration tools) to communicate with servers.

I was thinking of something more along the lines of allowing me to install desktop software, right there under one specific user account. Like either using an Ubuntu GUI or even another Windows OS install, and set it up on the server for that one account. Does this seem possible?

What I think you are asking for, is not possible, and I am not even sure why you would want it. Linux does not run Windows software unless you install an emulator like wine, and an emulator like that really should not be installed on a server that hosts Web sites for customers. It is going to use resources that could be used for hosting sites, making the sites slower.

cPanel effectively is its own GUI, so I am not exactly sure what you hope to accomplish. The GUI on Ubuntu is called X Window, and it should not be installed on a server, and in fact we do not support cPanel on servers that have it installed. cPanel gives you a Web-accessible GUI interface, so X Windows is effectively redundant, and that is part of why we do not support it, in addition to the fact that it uses server resources that can be better used to host sites.

Desktop applications should be run on a desktop running the operating system they were developed for, and server services should be run on a server.